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HOMOMORPHISMS BETWEEN MAPPING CLASS JAVIER ARAMAYONA & JUAN SOUTO
 

Summary: HOMOMORPHISMS BETWEEN MAPPING CLASS
GROUPS
JAVIER ARAMAYONA & JUAN SOUTO
Abstract. Suppose that X and Y are surfaces of finite topologi-
cal type, where X has genus g 6 and Y has genus at most 2g-1;
in addition, suppose that Y is not closed if it has genus 2g - 1.
Our main result asserts that every non-trivial homomorphism
Map(X) Map(Y ) is induced by an embedding, i.e. a combina-
tion of forgetting punctures, deleting boundary components and
subsurface embeddings. In particular, if X has no boundary then
every non-trivial endomorphism Map(X) Map(X) is in fact an
isomorphism.
As an application of our main theorem we obtain that, under
the same hypotheses on genus, if X and Y have finite analytic
type then every non-constant holomorphic map M(X) M(Y )
between the corresponding moduli spaces is a forgetful map. In
particular, there are no such holomorphic maps unless X and Y
have the same genus and Y has at most as many marked points as
X.
A nuestras madres, cada uno a la suya.

  

Source: Aramayona, Javier - Department of Mathematics, National University of Ireland, Galway

 

Collections: Mathematics